Try swapping the RCAs at the head unit. If it change sides, it's likely a head unit problem. You can measure the resistance from the RCA shield on the head unit to the case of the head unit. Both channels should read the same (generally 0 ohms).

Grounding the shield won't do any damage. Ground it before turning the system on. If the shield is open and you ground it while it's on, it could cause a significant transient (pop).

The manual mentions balanced operation for the RCA signal but doesn't give much detail. I would like to see the circuit so I can understand it better. If it was set up for balanced operation and you grounded the shield, it could cause the noise to get worse. Since you read ~0 ohms to ground, it's not likely set up for balanced operation. I don't know if the schematic diagram will help but it should eliminate some of the guesswork.